Question: What are the ingredients in the vaccines that have been the cause of severe allergic reactions for some people?

Answer: The short answer is that we don’t know what causes severe allergic reaction and scientists are actively researching it.  Here’s the longer answer: 

• Severe allergic reactions are rare and treatable. According to CDC’s vaccine adverse event reporting system, severe allergic reactions after Pfizer/Moderna vaccination have occurred in 2 to 5 people per every million vaccinated in the country (see JAMA paper).  Note: An allergic reaction is considered severe when a person needs to be treated with epinephrine or EpiPen or if the person must go to the hospital. Experts refer to severe allergic reactions as anaphylaxis.  
• If you have a known allergy to any ingredient in a COVID vaccine, CDC recommends against receiving that vaccine.  CDC’s guidance is here and for a full list of ingredients, see Pfizer, Moderna, and Johnson & Johnson fact sheets.  Of special note, if you are allergic to polyethylene glycol (PEG), you should not take Pfizer or Moderna vaccines.  And if you are allergic to polysorbate, you should not take the Johnson & Johnson vaccine.  
• Scientists hypothesize that PEG may be the culprit.  As described in a December 2020 report from Science, the Pfizer and Moderna vaccines use lipid nanoparticles (LNPs) to protect the mRNA; the LNPs are ““PEGylated”—chemically attached to PEG molecules that cover the outside of the particles and increase their stability and life span.”  PEG is used in medicines and everyday products like toothpaste and shampoo and were long thought to be biologically inert.  However, a growing body of evidence indicates that PEG is actually active.  A substantial proportion of people have antibodies to PEG. Those individuals with very high levels of PEG antibodies have experienced severe allergic reactions in other drug trials.  PEG allergy is also quite rare.
• The NIH is studying the mechanism of severe allergic reaction. According to clinicaltrials.gov, the NIH is about to begin recruiting 2040 individuals with a history of allergic reaction and 1360 individuals with no history of allergic reaction to participate in a randomized control trial to better understand systemic allergic reaction to both Pfizer and Moderna vaccines.  The study is expected to last 17 weeks, so hopefully we can see results in the summer.  While these results will get us closer to understanding the mechanism of reaction, more study will be required even after this trial.